Bracket words along Hardy field sequences

TL;DR

This paper investigates the properties of bracket words along Hardy field sequences, demonstrating their determinism and compliance with Sarnak's conjecture, thus extending understanding of sequence complexity and randomness.

Contribution

It introduces a novel analysis of bracket words along Hardy field sequences, showing their deterministic nature and confirming their alignment with Sarnak's conjecture.

Findings

Sequences are deterministic with sub-exponential subword complexity.

Sequences satisfy Sarnak's conjecture.

Generalizes properties of Sturmian and Piatetski-Shapiro sequences.

Abstract

We study bracket words, which are a far-reaching generalisation of Sturmian words, along Hardy field sequences, which are a far-reaching generalisation of Piatetski--Shapiro sequences $\lfloor n^c \rfloor$. We show that thus obtained sequences are deterministic (i.e., they have sub-exponential subword complexity) and satisfy Sarnak's conjecture.